***Appendix A***
Derivation of E and B Arbitrarily Close to an Oscillating Point Charge
G.R.Dixon, 7/14/2006
Define utility vector u to be
, (A1)
where D points from the charge’s retarded position to the field evaluation point r, and vR is the charge’s retarded velocity. If the field evaluation point is P=(0,dy,0), and if the motion is x=A Sin(wt), then at time t=0
, (A2a)
, (A2b)
, (A2c)
. (A2d)
The general solution for E(r,t) is
, (A3)
. (A4)
In the present case Eq. A3 simplifies to
. (A5)
The following figure depicts the direction of u (and E).
Figure A1
Direction of u (and E)
Referring to Fig. A1,
(A6)
, (A7)
, (A8)
. (A9)
Thus
. (A10)
And
, (A11)
. (A12)
Therefore
. (A13)
The Lorentz transformations for the field vector components are
, (A14)
. (A15)
For wA>v, Bz’ has the same sign as Bz and Sy’ is positive. If wA=v, then Bz’ and Sy are zero. And if wA<v (as in the present case), then Bz’ has the opposite sign of Bz and Sy’ is negative.